What are some examples of everyday systems that use AI?
A good, recent, and accessible book which includes many case studies is Prediction Machines. Check it out for more details than I can provide in this answer.

Example applications are all around us, but one of the problems with recognizing them is that the bar for what we call AI is constantly being raised.

Consider that a few decades ago, directions from google maps would certainly be recognized as AI, whereas now most laypeople wouldn't make that association.

Some other commonplace examples:


Amazon.com can guess at what you'd like to buy next.
Facebook and many other companies have programs that decide which ads to show you, without human intervention.
Tesla's autopilot feature will drive your car down a highway.
Google search can instantly return the results you were looking for, even when you enter strange or ambiguous search terms.
Virtual Assistants from many companies, like Alexa recognize your voice, figure out what words you're saying, and then figure out what you want them to do.
StackOverflow determines which questions are hot, and which ones need moderator attention.


There are are also a lot of less obvious examples. I'll pick a few from my research area:


When you spot a sky marshal on a plane, or a dog patrol in the airport it's because an AI system put them there. Some subway systems now use the same technology for fare checkers. 
When people trade agricultural products in Uganda, it's an AI system and AI techniques at work behind the scenes. 
When you're assigned a donor for a kidney transplant (and now other organs too), an AI system did the heavy lifting in that decision. 


Of course, there's lots more going on too, but those should give you some suggestions for everyday conversation.